本文主要探討Cosserat彈性理論的格林函數及其一階與二階微分，及其Eshelby張量。首先介紹Cosserat彈性理論的基本方程式，接著介紹格林函數的定義與概念，並使用傅立葉變換解Cosserat彈性材料的格林函數，在材料係數具有一定的對稱條件下，可以將格林函數簡化至一個單位圓的線積分，最後以高斯積分法作數值積分求出數值解並與中心對稱材料格林函數之解析解比較其誤差。之後推導格林函數的一階導數與二階導數，同樣可以簡化成一個單位圓的線積分，最後進行數值計算，並將結果與中心對稱且等向性之Cosserat材料的解析解相比較其誤差。最後推導Eshelby張量，並介紹當內含物為橢球形狀時的Eshelby張量。

Cosserat elasticity is a refined elasticity theory used to describe the deformation of elastic media with oriented particles. In addition to three displacements, a micropolar particle has an additional three micro-rotations, allowing it to link the couple moments as well as the stresses. The framework has an intrinsic length scale and is capable to characterize various materials with sufficient insight information, such as foams, materials with helical or twisted fibers. A more recent implication can be directed to the simulation of metametrials. Motivated by the latter progress, in this work, we aim to provide some theoretical studies on Cosserat elasticity. Specifically, we will derive the Green's function and its spatial derivatives. Inclusion problem in an infinite Cosserat medium is also examined. The method of Fourier transform is employed in the formulation. Centro- and non-centrosymmetry Cosserat elasticity are studied. We show that, under a certain restriction on the constitutive material parameters, Green's function can be expressed as a line integral along a unit circle. This form permits us to evaluate the results accurately by Gaussian quadrature. In addition, we find that, for the inclusion problems, even when the eigenstrain or the eigen-curvature is constant, the resulting strain and curvature tensor are not spatially uniform, in distinct with the famous constant Eshelby tensor in classical elasticity. In contrast it is a polynomial of position of second orders. We show that the resulting expressions can be expressed as a surface integral on a unit sphere and can be evaluated by numerical integration procedure. All these results permit us to make further explorations in metamaterials or in composites made of two or more different constituents.

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